Page 419 - Probability and Statistical Inference

396 8. Tests of Hypotheses

called simple provided that Θ , Θ are singleton subsets of Θ. That is, a
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hypothesis such as H : θ = θ or H : θ = θ , with θ ,θ known, would be
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called a simple hypothesis. A hypothesis which is not simple is called compos-
ite. A hypothesis such as H : θ > θ , H : θ < θ , H : θ ≥ θ , H : θ ≤ θ , for
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example, are referred to as one-sided composite hypotheses. A hypothesis
such as H : θ ≠ θ or H : θ ≤ θ ∪ θ ≥ θ with θ < θ , for example,
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is referred to as a two-sided composite hypothesis. In Section 8.2,
we first formulate the two types of errors in the decision making and focus
on the fundamental idea of a test. The Section 8.3 develops the concept of the
most powerful (MP) test for choosing between a simple null versus simple
alternative hypotheses. In Section 8.4, the idea of a uniformly most powerful
(UMP) test is pursued when H is simple but H is one-sided. The Section 8.5
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gives examples of two-sided alternative hypotheses testing situations and ex-
amines the possibilities of finding the UMP test. Section 8.5 touches upon the
ideas of unbiased and uniformly most powerful unbiased (UMPU) tests.
8.2 Error Probabilities and the Power Function
It is emphasized again that H and H are two statements regarding the un-
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known parameter θ. Then we gather around the random samples X , ..., X n
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from the appropriate population and learn about the unknown value of θ.
Intuitively speaking, the experimenter would favor H or H if the estimator
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seems more likely to be observed under H or H respec-
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tively. But, at the time of decision making, the experimenter may make mis-
takes by favoring the wrong hypothesis simply because the decision is based
on the evidence gathered from a random sample.
To understand the kinds of errors one can commit by choosing one hy-
pothesis over the other, the following table may be helpful. An examination
Table 8.2.1. Type I and Type II Errors
Test Result Nature’s Choice
or Decision H True H True
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Accept H No Error Type II Error
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Accept H Type I Error No Error
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of the entries reveals that the experimenter may commit one of the two
possible errors. One ends up rejecting the null hypothesis H while H is
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actually true or ends up accepting the null hypothesis H while H is true.
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Table 8.2.1 clearly suggests that the other two possible decisions are correct